Mastering the Algebra of Continuous Functions for Competitive Exams

Welcome to this module on the Algebra of Continuous Functions, a crucial topic for excelling in competitive mathematics exams like JEE. Understanding how basic arithmetic operations affect continuity is fundamental to solving complex calculus problems. We'll explore the rules governing the sum, difference, product, and quotient of continuous functions, along with the continuity of composite functions.

What are Continuous Functions?

A function $f(x)$ is continuous at a point $c$ if three conditions are met: 1. $f(c)$ is defined. 2. $\lim_{x \to c} f(x)$ exists. 3. $\lim_{x \to c} f(x) = f(c)$ . Intuitively, a continuous function can be drawn without lifting your pen from the paper. Polynomials, trigonometric functions, exponential functions, and logarithmic functions are continuous over their respective domains.

What are the three conditions for a function

f(x)

to be continuous at a point

c

$f(c)$ is defined. 2. $\lim_{x \to c} f(x)$ exists. 3. $\lim_{x \to c} f(x) = f(c)$ .

The Algebra of Continuous Functions

If two functions, $f(x)$ and $g(x)$ , are continuous at a point $c$ , then the following operations also result in functions continuous at $c$ :

Operation	Resulting Function	Condition
Sum	$f(x) + g(x)$	Continuous at $c$
Difference	$f(x) - g(x)$	Continuous at $c$
Product	$f(x) \cdot g(x)$	Continuous at $c$
Quotient	$\frac{f(x)}{g(x)}$	Continuous at $c$ , provided $g(c) \neq 0$

Remember the crucial condition for the quotient: the denominator must not be zero at the point of interest!

Continuity of Composite Functions

A key theorem states that if $g(x)$ is continuous at $c$ and $f(y)$ is continuous at $g(c)$ , then the composite function $(f \circ g)(x) = f(g(x))$ is continuous at $c$ . This means we can 'chain' continuous functions together, and the result remains continuous.

Consider two functions: $g(x) = x^2$ and $f(y) = \sin(y)$ . Both are continuous everywhere. For the composite function $f(g(x)) = \sin(x^2)$ , if we want to check continuity at $x=a$ , we first note that $g(x)$ is continuous at $a$ . Then, we evaluate $g(a) = a^2$ . Since $f(y) = \sin(y)$ is continuous at any value of $y$ , it is certainly continuous at $y = a^2$ . Therefore, by the composite function theorem, $\sin(x^2)$ is continuous at $x=a$ . This principle applies to any point $a$ . The visual below illustrates how the continuity 'propagates' through the composition.

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Text-based content

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g(x)

is continuous at

c

and

f(y)

is continuous at

g(c)

, what can we say about the composite function

f(g(x))

c

The composite function $f(g(x))$ is continuous at $c$ .

Practical Application in Exams

In competitive exams, you'll often encounter functions that are combinations of basic continuous functions. For example, a function like $h(x) = \frac{\sin(x)}{x^2 + 1}$ is continuous wherever its denominator is non-zero. Since $x^2 + 1$ is always positive, $h(x)$ is continuous for all real numbers. Similarly, functions involving piecewise definitions often require checking continuity at the boundary points using these algebraic rules and the definition of continuity.

Summary of Key Takeaways

The sum, difference, and product of continuous functions are continuous. The quotient is continuous provided the denominator is non-zero. Composite functions formed by continuous functions are also continuous. Mastering these rules will simplify your approach to continuity problems in calculus.

Learning Resources

Continuity of Functions - Definition and Properties(documentation)

Provides a clear definition of continuity and outlines the algebraic properties of continuous functions with examples.

Continuity and Differentiability - Algebra of Derivatives(video)

While focused on derivatives, this video explains the continuity rules that underpin them, offering a good foundational understanding.

Continuity of Composite Functions(documentation)

A detailed explanation of the theorem regarding the continuity of composite functions with illustrative examples.

JEE Mathematics: Continuity and Differentiability(blog)

A comprehensive overview of continuity and differentiability tailored for JEE preparation, including algebraic properties.

Properties of Continuous Functions(wikipedia)

A discussion on StackExchange covering various properties of continuous functions, including their behavior under arithmetic operations.

Calculus I - Continuity(documentation)

An in-depth guide to continuity, including the algebra of continuous functions and examples relevant to calculus problems.

Continuity of Functions - Examples and Practice(blog)

Offers practical examples and practice problems related to the continuity of functions, including those involving algebraic operations.

Understanding Composite Functions(video)

Focuses on the composition of functions, which is essential for understanding the continuity of composite functions.

Continuity and Differentiability for JEE Main & Advanced(blog)

A resource specifically aimed at JEE preparation, covering the essential concepts of continuity and differentiability.

Theorems on Continuity(wikipedia)

A forum discussing important theorems related to continuity, including the algebra of continuous functions and composite functions.